reserve x,y,z,X,Y for set;
reserve X,Y for non empty set,
  f for Function of X,Y;
reserve X, Y for non empty set,
  F for (BinOp of Y),
  B for (Element of Fin X),
  f for Function of X,Y;

theorem
  F is commutative & F is associative & F is idempotent & F is
  having_a_unity implies for f,g being Function of X,Y for A,B being Element of
  Fin X st f.:A = g.:B holds F$$(A,f) = F$$(B,g)
proof
  assume that
A1: F is commutative & F is associative and
A2: F is idempotent and
A3: F is having_a_unity;
  let f,g be Function of X,Y;
  let A,B be Element of Fin X such that
A4: f.:A = g.:B;
  now
    assume
A5: A = {};
    then A = {}.X;
    then
A6: F$$(A,f) = the_unity_wrt F by A1,A3,Th28;
    f.:A = {} by A5;
    then B = {}.X by A4,Th10;
    hence thesis by A1,A3,A6,Th28;
  end;
  hence thesis by A1,A2,A4,Th23;
end;
